## Orderings for incomplete factorization preconditioning of nonsymmetric problems (1999)

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Venue: | SIAM J. SCI. COMPUT |

Citations: | 51 - 11 self |

### BibTeX

@ARTICLE{Benzi99orderingsfor,

author = {Michele Benzi and Daniel B. Szyld and Arno Van Duin},

title = {Orderings for incomplete factorization preconditioning of nonsymmetric problems},

journal = {SIAM J. SCI. COMPUT},

year = {1999},

volume = {20},

number = {5},

pages = {1652--1670}

}

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### Abstract

Numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear systems is shown. The preconditioners used in this study are different variants of incomplete factorizations. It is shown that certain reorderings for direct methods, such as reverse Cuthill–McKee, can be very beneficial. The benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity.

### Citations

1673 |
Iterative Methods for Sparse Linear Systems
- SAAD
- 2003
(Show Context)
Citation Context ...for nonsymmetric systems, including GMRES [43], Bi-CGSTAB [48], and transpose-free QMR (TFQMR) [28]; for a description of these, as well as a description of incomplete factorizations, see, e.g., [3], =-=[42]-=-. In this paper, we mainly concentrate on orderings originally devised for matrix factorizations, i.e., those used to reduce fill-in in the factors; see, e.g., [18] or [29]. We want to call attention ... |

1489 | GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
- Saad, Schultz
- 1986
(Show Context)
Citation Context ...gs are based on the (undirected) graph associated with the structure of A; otherwise, the structure of A + A T is used. We consider several iterative methods for nonsymmetric systems, including GMRES =-=[43]-=-, Bi-CGSTAB [48], and transpose-free QMR (TFQMR) [28]; for a description of these, as well as a description of incomplete factorizations, see, e.g., [3], [42]. In this paper, we mainly concentrate on ... |

573 |
Direct Method for Sparse Matrices
- Duff, Erisman, et al.
- 1989
(Show Context)
Citation Context ...ete factorizations, see, e.g., [3], [42]. In this paper, we mainly concentrate on orderings originally devised for matrix factorizations, i.e., those used to reduce fill-in in the factors; see, e.g., =-=[18]-=- or [29]. We want to call attention to the fact that a permutation of the variables (and equations) # Received by the editors September 4, 1997; accepted for publication (in revised form) March 31, 19... |

534 |
Computer Solution of Large Sparse Positive Definite Matrices
- GEORGE, LIU
- 1981
(Show Context)
Citation Context ...orizations, see, e.g., [3], [42]. In this paper, we mainly concentrate on orderings originally devised for matrix factorizations, i.e., those used to reduce fill-in in the factors; see, e.g., [18] or =-=[29]-=-. We want to call attention to the fact that a permutation of the variables (and equations) ∗ Received by the editors September 4, 1997; accepted for publication (in revised form) March 31, 1998; publ... |

529 |
der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Society for Industrial and Applied Mathematics
- Barrett, Berry, et al.
- 1994
(Show Context)
Citation Context ...hods for nonsymmetric systems, including GMRES [43], Bi-CGSTAB [48], and transpose-free QMR (TFQMR) [28]; for a description of these, as well as a description of incomplete factorizations, see, e.g., =-=[3]-=-, [42]. In this paper, we mainly concentrate on orderings originally devised for matrix factorizations, i.e., those used to reduce fill-in in the factors; see, e.g., [18] or [29]. We want to call atte... |

361 | Iterative Solution of Large Linear System - Young - 1971 |

355 |
der Vorst. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems
- van
- 1992
(Show Context)
Citation Context ...the (undirected) graph associated with the structure of A; otherwise, the structure of A + A T is used. We consider several iterative methods for nonsymmetric systems, including GMRES [43], Bi-CGSTAB =-=[48]-=-, and transpose-free QMR (TFQMR) [28]; for a description of these, as well as a description of incomplete factorizations, see, e.g., [3], [42]. In this paper, we mainly concentrate on orderings origin... |

294 |
Sparse matrix test problems
- Duff, Grimes, et al.
- 1989
(Show Context)
Citation Context ...s can be applied to other problems. For this reason, we discuss additional experiments performed on a selection of nonsymmetric matrices from various sources, including the Harwell--Boeing collection =-=[19]-=- and Saad's SPARSKIT [40]. These matrices arise from di#erent application areas: oil reservoir modeling, plasma physics, neutron di#usion, metal forming simulation, etc. Some of these matrices arise f... |

273 | SPARSEKIT: A Basic Tools Kit for Sparse Matrix Computations
- Saad
- 1994
(Show Context)
Citation Context ... problems. For this reason, we discuss additional experiments performed on a selection of nonsymmetric matrices from various sources, including the Harwell--Boeing collection [19] and Saad's SPARSKIT =-=[40]-=-. These matrices arise from di#erent application areas: oil reservoir modeling, plasma physics, neutron di#usion, metal forming simulation, etc. Some of these matrices arise from finite element modeli... |

268 |
der Vorst, An iterative solution method for linear systems of which the coefficient matrix is symmetric M-matrix
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- 1977
(Show Context)
Citation Context ...the following accelerators: Bi-CGSTAB, TFQMR, and GMRES with restart parameter m = 20. The preconditioners used were standard incomplete factorizations based on levels of fill (ILU(0) and ILU(1); see =-=[35]-=-) and Saad's dual threshold ILUT; see [41], [42]. For ILUT, we used two di#erent sets of parameters, (10 -2 , 5) and (10 -3 , 10). The latter results in a very powerful but expensive preconditioner, c... |

144 |
Modification of the minimum degree algorithm by multiple elimination
- LIU
- 1985
(Show Context)
Citation Context ...e to the departure from symmetry than others. There are many possible ways of implementing the (reverse) Cuthill--McKee and minimum degree reorderings. We used Liu's multiple minimum degree algorithm =-=[34]-=-, and for the Cuthill--McKee reorderings, we used an implementation which chooses a pseudoperipheral node as starting node and sorts nodes in the same level set by increasing degree [29]. Other strate... |

143 |
A transpose-free quasi-minimal residual algorithm for non-hermitian linear systems
- Freund
- 1993
(Show Context)
Citation Context ...h the structure of A; otherwise, the structure of A + A T is used. We consider several iterative methods for nonsymmetric systems, including GMRES [43], Bi-CGSTAB [48], and transpose-free QMR (TFQMR) =-=[28]-=-; for a description of these, as well as a description of incomplete factorizations, see, e.g., [3], [42]. In this paper, we mainly concentrate on orderings originally devised for matrix factorization... |

91 | ILUT: A dual threshold incomplete LU factorization
- Saad
- 1992
(Show Context)
Citation Context ...MR, and GMRES with restart parameter m = 20. The preconditioners used were standard incomplete factorizations based on levels of fill (ILU(0) and ILU(1); see [35]) and Saad's dual threshold ILUT; see =-=[41]-=-, [42]. For ILUT, we used two di#erent sets of parameters, (10 -2 , 5) and (10 -3 , 10). The latter results in a very powerful but expensive preconditioner, containing up to five times the number of n... |

74 | The design and use of algorithms for permuting large entries to the diagonal of sparse matrices
- Duff, Koster
- 1999
(Show Context)
Citation Context ...onal. This may cause trouble for the construction of ILU preconditioners. Therefore, the matrix was first reordered into a form with a zerofree diagonal, using a nonsymmetric permutation described in =-=[20]-=-. As for problem kershaw60x60, this matrix was extracted from the AUGUSTUS unstructured mesh di#usion package developed by Michael Hall at Los Alamos National Laboratory; see [32]. It should be mentio... |

62 | Experimental study of ILU preconditioners for indefinite matrices
- Chow, Saad
- 1997
(Show Context)
Citation Context ...e far from being diagonally dominant, the norm of the residual matrix R alone is usually not a reliable indicator of the quality of the corresponding preconditioner. It has been pointed out, e.g., in =-=[11]-=-, that a more revealing measure of the quality of the preconditioner can be obtained by considering the Frobenius norm of the deviation of the preconditioned matrix from the identity, i.e., #I - A(sLs... |

56 |
Ordering methods for preconditioned conjugate gradient methods applied to unstructured grid problems
- D'Azevedo, Forsyth, et al.
- 1992
(Show Context)
Citation Context ...fluence of reorderings on the convergence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], =-=[14]-=-, [21], [24], [27], [36], [37], [38], [45], [49]. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradients, i.e., ... |

31 | The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations - Dutto - 1993 |

30 |
Preconditioning techniques for non-symmetric and indefinite linear systems
- Saad
- 1988
(Show Context)
Citation Context ...quare# = (0, 1) (0, 1) : -##u + #e xy u #x + #e -xy u #y = g (1) with homogeneous Dirichlet boundary conditions. Equation (1) has been repeatedly used as a model problem in the literature; see, e.g., =-=[39]-=-. The problem is discretized using centered di#erences for both the second order and first order derivatives with 1656 MICHELE BENZI, DANIEL B. SZYLD, AND ARNO VAN DUIN grid size h = 1/33, leading to ... |

26 | Analysis of parallel incomplete point factorizations, Linear Algebra and its Application
- Eijkhout
- 1991
(Show Context)
Citation Context ...eorderings on the convergence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], [14], [21], =-=[24]-=-, [27], [36], [37], [38], [45], [49]. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradients, i.e., in the symme... |

25 | Application of threshold partitioning of sparse matrices to Markov chains
- Choi, Szyld
- 1996
(Show Context)
Citation Context ...xpensive to be practical, except for rather simple problems. In addition to reorderings which were originally designed for direct methods, one can use the permutation produced by the algorithm TPABLO =-=[10]-=-, which also uses the magnitude of the entries of the matrix. This algorithm produces a permuted matrix with dense diagonal blocks, while the entries outside the blocks on the diagonal have magnitude ... |

25 |
Meurant, The effect of orderings on preconditioned conjugate gradients
- Duff, A
- 1989
(Show Context)
Citation Context ...e of reorderings on the convergence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], [14], =-=[21]-=-, [24], [27], [36], [37], [38], [45], [49]. In this context, Duff and Meurant [21] have performed a very detailed study of the effects of reorderings for preconditioned conjugate gradients, i.e., in t... |

22 | Threshold ordering for preconditioning nonsymmetric problems
- Benzi, Choi, et al.
- 1997
(Show Context)
Citation Context ...reatment of certain Markov chain problems [10]; see also [23]. It turns out that this reordering is also useful for point incomplete factorizations, where it is often better than the natural ordering =-=[6]-=- and some of the reorderings considered in this paper [7]. However, for most cases treated in this paper, the performance of TPABLO is inferior to that of some of the reorderings designed for direct m... |

22 | Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems - Elman - 1989 |

20 |
A Portable MPI Implementation of the SPAI Preconditioner in ISIS
- Barnard, Clay
- 1997
(Show Context)
Citation Context ...ix and NZ is the number of nonzeros. The degree of di#culty of these problems varies from moderate (watt2) to extreme (utm5940 and fidap007). Concerning problem ale1590, which was provided by Barnard =-=[2]-=-, the original ordering caused the coe#cient matrix to have some zero entries on the main diagonal. This may cause trouble for the construction of ILU preconditioners. Therefore, the matrix was first ... |

19 |
der Vorst, ‘Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from pde-problems
- van
- 1981
(Show Context)
Citation Context ...in in the factors, e.g., using ILU(1) or ILUT instead of ILU(0), will solve the problem in some cases but not always. This kind of instability of ILU factorizations was first noticed by van der Vorst =-=[47]-=- and analyzed in detail by Elman [25]. We will see that in some cases, reordering the coe#cient matrix before performing the incomplete factorization can have the e#ect of producing stable triangular ... |

18 |
Ordering techniques for the preconditioned conjugate gradient method on parallel computers
- Elman, Agrón
- 1989
(Show Context)
Citation Context ...ings on the convergence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], [14], [21], [24], =-=[27]-=-, [36], [37], [38], [45], [49]. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradients, i.e., in the symmetric p... |

17 |
On parallelism and convergence of incomplete LU factorizations
- Doi
- 1991
(Show Context)
Citation Context ... lower bound for #(sLsU) -1 ## but is quite useful in practice. The e#ects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in [9], [13], [15], =-=[16]-=-, [17], [22], [33], [44], [46]. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods; see, e.g., [14], ... |

16 |
A stability analysis of incomplete
- Elman
- 1986
(Show Context)
Citation Context ... or ILUT instead of ILU(0), will solve the problem in some cases but not always. This kind of instability of ILU factorizations was first noticed by van der Vorst [47] and analyzed in detail by Elman =-=[25]-=-. We will see that in some cases, reordering the coe#cient matrix before performing the incomplete factorization can have the e#ect of producing stable triangular factors, and hence more e#ective prec... |

15 |
Parallel elliptic preconditioners: Fourier analysis and performance on the Connection Machine
- Chan, Kuo, et al.
- 1989
(Show Context)
Citation Context ...ure. The influence of reorderings on the convergence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only =-=[8]-=-, [12], [14], [21], [24], [27], [36], [37], [38], [45], [49]. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradi... |

15 |
Don’t suppress the wiggles-they’re telling you something’, Cornput. Fluids, 9,223
- Gresho, Lee
- 1981
(Show Context)
Citation Context ...e [30]), and, moreover, such approximations are only first order accurate and in many cases are unable to resolve fine features of the solution, such as boundary layers. In this case, as suggested in =-=[31], a unifor-=-m coarse grid could be used to determine the region where the boundary layer is located (this corresponds to "wiggles" in an otherwise smooth solution). This would require solving linear sys... |

14 |
Comparison of fast iterative methods for symmetric systems
- Behie, Forsyth
- 1983
(Show Context)
Citation Context ...uthill--McKee and some variants of it were found to perform satisfactorily for symmetric, strongly anisotropic problems due to the fact that these orderings are relatively insensitive to anisotropies =-=[4]-=-, [12], [49]. Numerical experiments indicating that the D2 diagonal ordering (which is a special case of Cuthill--McKee) for ILU(k) preconditioning of certain nonsymmetric problems defined on rectangu... |

13 | Weighted graph based ordering techniques for preconditioned conjugate gradient methods - Clift, Tang - 1995 |

13 | SOR as a preconditioner
- DeLong, Ortega
- 1995
(Show Context)
Citation Context ...only a lower bound for #(sLsU) -1 ## but is quite useful in practice. The e#ects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in [9], [13], =-=[15]-=-, [16], [17], [22], [33], [44], [46]. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods; see, e.g., ... |

12 |
Incomplete factorization methods for fully implicit simulation of enhanced oil recovery
- Behie, Forsyth
- 1984
(Show Context)
Citation Context ... ordering (which is a special case of Cuthill--McKee) for ILU(k) preconditioning of certain nonsymmetric problems defined on rectangular grids can be superior to the natural ordering were reported in =-=[5]-=-, but this observation did not receive the attention it deserved. This may be due in part to the fact that the authors use a terminology that is peculiar to the field of reservoir simulation. A short ... |

12 |
Conjugate Gradient Methods and ILU Preconditioning of Non-Symmetric Matrix Systems with Arbitrary Sparsity Patterns,” Int
- Langtangen
- 1989
(Show Context)
Citation Context ...(sLsU) -1 ## but is quite useful in practice. The e#ects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in [9], [13], [15], [16], [17], [22], =-=[33]-=-, [44], [46]. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods; see, e.g., [14], [33], [44]. Simon ... |

12 |
Orderings for conjugate gradient preconditionings
- Ortega
- 1991
(Show Context)
Citation Context ...convergence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], [14], [21], [24], [27], [36], =-=[37]-=-, [38], [45], [49]. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradients, i.e., in the symmetric positive defi... |

12 |
Multicolor ICCG methods for vector computers
- Poole, Ortega
- 1987
(Show Context)
Citation Context ...gence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], [14], [21], [24], [27], [36], [37], =-=[38]-=-, [45], [49]. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradients, i.e., in the symmetric positive definite c... |

11 |
On preconditioned Krylov subspace methods for discrete convection-diffusion problems
- Chernesky
- 1997
(Show Context)
Citation Context ...s. This is only a lower bound for #(sLsU) -1 ## but is quite useful in practice. The e#ects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in =-=[9]-=-, [13], [15], [16], [17], [22], [33], [44], [46]. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods;... |

11 |
A Conjugate Gradient Truncated Direct Method for the Iterative Solution of the Reservoir Simulation Pressure Equation
- Watts
(Show Context)
Citation Context ...conditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], [14], [21], [24], [27], [36], [37], [38], [45], =-=[49]-=-. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradients, i.e., in the symmetric positive definite case. Based o... |

11 |
A Graph-Theory Approach for Analyzing the Effects of Ordering on
- Doi, Lichnewsky
- 1991
(Show Context)
Citation Context ... bound for �( ¯ LŪ)−1�∞ but is quite useful in practice. The effects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in [9], [13], [15], [16], =-=[17]-=-, [22], [33], [44], [46]. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods; see, e.g., [14], [33], ... |

10 |
Vectorizable preconditioners for elliptic difference equations in three space dimensions
- Eijkhout
- 1989
(Show Context)
Citation Context ...in the incomplete factorization (as conjectured by Simon [44]) but is almost directly related to the norm of the residual matrix R = A − ¯ L¯ LT , where ¯ L is an incomplete Cholesky factor of A; see =-=[1]-=- for a rigorous derivation of this result under appropriate conditions. Throughout the paper we abuse the notation and use A to denote both the original matrix and the permuted one. Similarly, ¯ L and... |

9 |
Meurant, The e#ect of ordering on preconditioned conjugate gradients
- Duff, A
- 1989
(Show Context)
Citation Context ...e of reorderings on the convergence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], [14], =-=[21]-=-, [24], [27], [36], [37], [38], [45], [49]. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradients, i.e., in the... |

7 |
Parallelizable block diagonal preconditioners for the compressible Navier-Stokes equations
- Dutto, Habashi, et al.
- 1994
(Show Context)
Citation Context ...umerical values. The original motivation for the TPABLO algorithm was to produce good block diagonal preconditioners, and also blocks for the treatment of certain Markov chain problems [10]; see also =-=[23]-=-. It turns out that this reordering is also useful for point incomplete factorizations, where it is often better than the natural ordering [6] and some of the reorderings considered in this paper [7].... |

6 |
Incomplete LU preconditioners for conjugate-gradient-type iterative methods
- Simon
- 1985
(Show Context)
Citation Context ...ering. An important observation in [21] is that the number of conjugate gradient iterations is not related to the number of fill-ins discarded in the incomplete factorization (as conjectured by Simon =-=[44]-=-) but is almost directly related to the norm of the residual matrix R = A -sLsL T , wheresL is an incomplete Cholesky factor of A; see [1] for a rigorous derivation of this result under appropriate co... |

6 |
Orderings for parallel conjugate gradient preconditioners
- Stotland, Ortega
- 1997
(Show Context)
Citation Context ...of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [8], [12], [14], [21], [24], [27], [36], [37], [38], =-=[45]-=-, [49]. In this context, Du# and Meurant [21] have performed a very detailed study of the e#ects of reorderings for preconditioned conjugate gradients, i.e., in the symmetric positive definite case. B... |

5 | Diagonal dominance and positive definiteness of upwind approximations for advection diffusion problems
- Golub, Silvester, et al.
- 1996
(Show Context)
Citation Context ...rom this problem and give rise to matrices with very nice properties from the point of view of iterative solutions, such as diagonal dominance. However, this may not be true for nonuniform grids (see =-=[30]-=-), and, moreover, such approximations are only first order accurate and in many cases are unable to resolve fine features of the solution, such as boundary layers. In this case, as suggested in [31], ... |

2 |
An automatic ordering method for incomplete factorization iterative solvers
- D’Azevedo, Forsyth, et al.
- 1991
(Show Context)
Citation Context ...is is only a lower bound for #(sLsU) -1 ## but is quite useful in practice. The e#ects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in [9], =-=[13]-=-, [15], [16], [17], [22], [33], [44], [46]. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods; see, ... |

2 |
A Graph-Theory Approach for Analyzing the E#ects of Ordering on
- Doi, Lichnewsky
- 1991
(Show Context)
Citation Context ... bound for #(sLsU) -1 ## but is quite useful in practice. The e#ects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in [9], [13], [15], [16], =-=[17]-=-, [22], [33], [44], [46]. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods; see, e.g., [14], [33], ... |

2 |
Contribution to Biorthogonalization Algorithms for the Solution of Linear Equations
- Sarmiento
- 1995
(Show Context)
Citation Context ...but is quite useful in practice. The effects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in [9], [13], [15], [16], [17], [22], [33], [44], =-=[46]-=-. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods; see, e.g., [14], [33], [44]. Simon [44] used qu... |

1 |
Vectorizable preconditioners for elliptic di#erence equations in three space dimensions
- Axelsson, Eijkhout
- 1991
(Show Context)
Citation Context ...ded in the incomplete factorization (as conjectured by Simon [44]) but is almost directly related to the norm of the residual matrix R = A -sLsL T , wheresL is an incomplete Cholesky factor of A; see =-=[1]-=- for a rigorous derivation of this result under appropriate conditions. Throughout the paper we abuse the notation and use A to denote both the original matrix and the permuted one. Similarly,sL andsU... |